Foundations of Data Science

Setting λ = ln(1 + δ)
Prob ( s > (1 + δ)m ) ≤ ( e − ln(1+δ)) (1+δ)m
∏
n e p(eln(1+δ) −1)
i=1
( ) (1+δ)m 1 ∏ n
≤
1 + δ
i=1
( ) (1+δ)m 1
≤
e npδ
(1 + δ)
( ) m
e δ
≤
.
(1 + δ) (1+δ)
e pδ
To simplify the bound of Theorem 12.3, observe that
Therefore
(1 + δ) ln (1 + δ) = δ + δ2
2 − δ3
6 + δ4
12 − · · · .
(1 + δ) (1+δ) = e δ+ δ2 2 − δ3 6 + δ4
12 −···
and hence
e δ
(1+δ) (1+δ)
= e − δ2 2 + δ3 6 −···.
Thus, the bound simplifies to
Prob ( s < (1 + δ) m ) ≤ e − δ2 2 m+ δ3 6 m−···.
For small δ the probability drops exponentially with δ 2 .
When δ is large another simplification is possible. First
Prob ( s > (1 + δ) m ) ( ) m ( )
e δ
(1+δ)m e
≤
≤
(1 + δ) (1+δ) 1 + δ
If δ > 2e − 1, substituting 2e − 1 for δ in the denominator yields
Prob(s > (1 + δ) m) ≤ 2 −(1+δ)m .
Theorem 12.3 gives a bound on the probability of the sum being greater than the
mean. We now bound the probability that the sum will be less than its mean.
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